We quantify the luminosity contribution of active galactic nuclei (AGN) to the 12 um, mid-infrared (MIR; 5-38 um), and total IR (5-1000 um) emission in the local AGN detected in the all-sky 70 month Swift/BAT ultrahard X-ray survey. We decompose the IR spectral energy distributions (SEDs) of 587 objects into the AGN and starburst components using templates for an AGN torus and a star-forming galaxy. This enables us to recover the emission from the AGN torus including the low-luminosity end, down to log L(14-150keV)~41, which typically has significant host galaxy contamination. The total IR AGN luminosity obtained through the IR SED decomposition enables us to estimate the fraction of the sky obscured by dust, i.e., the dust covering factor. We demonstrate that the median dust covering factor is always smaller than the median X-ray obscuration fraction above an AGN bolometric luminosity of log Lbol(AGN)~42.5. Considering that the X-ray obscuration fraction is equivalent to the covering factor coming from both the dust and gas, this indicates that an additional neutral gas component, along with the dusty torus, is responsible for the absorption of X-ray emission. We also find ""pure-IR"" AGN whose far-IR emission is likely to be dominated by AGN component. Their AGN properties suggest that the dominating AGN contribution to the total IR band is not related to their higher AGN luminosities, lower BH masses, or higher Eddington ratio, while it implies that they have weaker star-formation luminosities than other AGN with similar luminosities. Considering this, their host galaxies might have already experienced AGN feedback in the past while their AGN luminosity is still active.